WorkiNg PaPer

Fire sales, indirect contagion and systemic stress testing

Norges BaNk research

2 | 2017

Rama Cont andERiC SChaanning

WorkiNg PaPer

Norges BaNk Working PaPer xx | 2014

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ISSN 1502-819-0 (online) ISBN 978-82-7553-963-0 (online)

Fire sales, indirect contagion and systemic stress testing

Rama Cont Eric Schaanning

Imperial College London Norges Bank Research March 17, 2017

Abstract

We present a framework for quantifying the impact of fire sales in a network of financial institutions with common asset holdings, subject to leverage or capital constraints. Asset losses triggered by macro-shocks may interact with one-sided portfolio constraints, such as leverage or capital constraints, resulting in liquidation of assets, which in turn affects market prices, leading to contagion of losses and possibly new rounds of fire sales when portfolios are marked to market.

Price-mediated contagion occurs through common asset holdings, which we quantify through liquidity-weighted overlaps across portfolios. Exposure to price- mediated contagion leads to the concept of indirect exposure to an asset class, as a consequence of which the risk of a portfolio depends on the matrix of asset holdings of other large and leveraged portfolios with similar assets.

Our model provides an operational stress testing method for quantifying the systemic risk arising from these effects. Using data from the European Banking Authority, we examine the exposure of the EU banking system to price-mediated contagion. Our results indicate that, even with optimistic estimates of market depth, moderately large macro-shocks may trigger fire sales which may then lead to substantial losses across bank portfolios, modifying the outcome of bank stress tests.

Price-mediated contagion leads to a heterogeneous cross-sectional loss distribution across banks, which cannot be replicated simply by applying a macro-shock to bank portfolios in absence of fire sales.

Unlike models based on ‘leverage targeting’, which assume symmetric reactions to gains or losses, our approach is based on the asymmetric interaction of portfolio losses with one-sided constraints, distinguishes between insolvency and illiquidity and leads to substantially different loss estimates in stress scenarios.

∗This working paper was prepared as part of the authors’ work at Norges Bank Research. The views expressed are those of the authors and do not necessarily reflect those of Norges Bank and at the Isaac Newton Institute for Mathematical Sciences, Cambridge, during the programme onSystemic Risk: Math- ematical modeling and Interdisciplinary Approaches, supported by EPSRC grant no EP/K032208/1. We thank Farooq Akram, Tobias Adrian, Laurent Clerc, Fernando Duarte, Darrell Duffie, Thomas Eisenbach, Martin Hellwig, Anil Kashyap, Andrei Kirilenko, David Levermore, Barbara Meller, David Murphy, Ser- gio Nicoletti-Altimari, Martin Summer and Lakshithe Wagalath for stimulating discussions on this topic.

Eric Schaanning is grateful for funding from the Fonds National de la Recherche Luxembourg (FNR) for his PhD studies under the AFR Grant scheme. We are grateful for the helpful and constructive suggestions of an anonymous referee that improved the presentation of our results.

Contents

1 Introduction 3

1.1 The need for macroprudential stress tests . . . 3

1.2 Summary and main findings . . . 5

1.3 Outline . . . 6

2 Modeling spillover effects from fire sales 7 2.1 Balance sheets and portfolio constraints . . . 7

2.2 Stress scenarios . . . 8

2.3 Deleveraging . . . 9

2.4 Market impact and price-mediated contagion . . . 11

2.5 Feedback loops, insolvency and illiquidity . . . 14

2.6 Comparison with “leverage targeting” . . . 15

3 A systemic stress test of the European banking system 16 3.1 Data . . . 16

3.2 Market impact and market depth . . . 18

3.3 Portfolio overlaps . . . 21

3.4 Stress scenarios . . . 24

3.5 Systemic stress test of European banks: results . . . 26

4 Indirect exposures 33 4.1 Notional vs effective exposures . . . 33

4.2 Indirect exposures: empirical evidence . . . 35

4.3 Relevance of indirect contagion for bank stress tests . . . 38

4.4 Why indirect exposures cannot be reproduced in single-bank stress tests . . 41

5 Implications for macroprudential stress testing and regulation 43 A Appendix 48 A.1 Sources for market data . . . 48

A.2 EBA: data identifiers and residual exposures . . . 49

1 Introduction

1.1 The need for macroprudential stress tests

In October 2008, the IMF’s estimate of losses in the US subprime mortgage sector was on the order of 500 bn USD, a large loss but still lower than, say, the loss from the Dot- com bubble burst Hellwig (2009). However, as the US subprime crisis developed into a full-blown global financial crisis, losses spilled over into other asset classes, sectors and countries and ballooned into trillions of dollars Hellwig (2009), exceeding the losses of the Dot-com bubble by an order of magnitude.

Supervisory stress tests for banks, which have become a cornerstone of financial reg- ulation, have focused on examining the resilience of bank balance sheets to severe stress scenarios. However, as the above example illustrates, loss amplification mechanisms which may compound initial losses may be just as important for understanding the nature of systemic risk.

Indeed, as pointed out in the Basel Committee on Banking Supervision’s recent report, stress tests conducted by bank supervisors still lack a genuine macroprudential component Basel Committee on Banking Supervision (2015). The report identifies the key missing ingredients as “endogenous reactions and feedback effects to initial stress”. As noted by ECB Vice-President Vitor Constˆancio Anderson (2016), in the current approach to bank stress tests “no bank reaction is considered. It would be far more realistic to assume that market participants could react to adverse conditions, rather than assuming passive bank behaviour throughout the entire stress test period. Bank behaviour or reaction could take the form of deleveraging, straight capital increases or working out of non-performing loans.”

There is ample empirical evidence that such deleveraging occurred on a large scale in 2008, leading to cross-asset contagion and amplification of losses in the financial system Kashyap et al. (2008); Brunnermeier and Pedersen (2009); Khandani and Lo (2011);

Manconi et al. (2012); Cont and Wagalath (2016). The well-documented occurrence of fire sales during market downturns Ellul et al. (2011); Coval and Stafford (2007); Shleifer and Vishny (2011); Jotikasthira et al. (2012) is not a coincidence: portfolio constraints -capital, leverage or liquidity constraints- that financial institutions are subject to forces them to deleverage when these constraints are breached as a result of losses, leading to fire sales of assets Kyle and Xiong (2001); Cont and Wagalath (2013). Similar large- scale deleveraging is also foreseeable in future stress scenarios, and foreseen by financial institutions themselves: “If we are unable to raise needed funds in the capital markets (including through offerings of equity and regulatory capital securities), we may need to liquidate unencumbered assets to meet our liabilities. In a time of reduced liquidity, we may be unable to sell some of our assets, or we may need to sell assets at depressed prices, which in either case could adversely affect our results of operations and financial condition” Credit Suisse (2015).

Fire sales generate endogenous risk Shin (2010) and can act as channel of loss conta- gion across asset classes and across financial institutions holding these assets Cont and Wagalath (2016); Caccioli et al. (2014). Unlike direct contagion through counterparty ex- posures Cont et al. (2013), fire-sales spillovers are mediated by prices and thus defy limits on counterparty exposures and institutional ring-fencing measures. As noted by Glasser- man and Young (2014), following the introduction of large exposure limits and collateral requirements, the likelihood of direct contagion through counterparty exposures has di- minished in the banking system.

It is therefore important for supervisors to include in macro-stress testing frameworks used for assessing bank capital adequacy the impact of fire sales and the deleveraging of portfolios in stress scenarios. This is especially relevant given that, post-crisis, supervisory stress tests have set a binding constraint for bank capital adequacy.¹

Fire sales and the resulting destabilizing feedback effects have been extensively studied in the literature Kyle and Xiong (2001); Shleifer and Vishny (2011) from a conceptual viewpoint. The challenge is to develop a quantitative framework versatile enough to be taken to empirical data and used in an operational macro-stress testing framework to quantify the endogenous risk and spillover effects arising from fire sales.

The goal of the present work is to address this challenge, by proposing a modeling framework for quantifying the exposure of the financial system to the endogenous losses and feedback effects resulting from fire sales in a macro-stress scenario. We provide a detailed discussion of the model, its use for the design of systemic stress tests, and the results obtained by applying the methodology to EU bank portfolios.

Previous attempts by regulators to account indirectly for the impact of fire sales in bank stress tests include the use of (exogenously specified) liquidation costs or increasing the severity of shocks in single-bank stress tests to account for possible loss amplification due to feedback from fire sales. These adjustments may mimick the severity of potential losses which may result from fire sales but fail to capture key cross-sectional features of fire-sales spillovers, such as the contagion across asset classes and the heterogeneous distribution of fire-sales losses across financial institutions and across asset classes of varying liquidity.

More recently, Greenwood et al. (2015); Duarte and Eisenbach (2013) have proposed a stress testing approach incorporating the impact on asset prices of deleveraging in bank portfolios, based on the assumption of leverage-targeting Adrian and Shin (2010) i.e.

that, in response to a shock, financial institutions rebalance their portfolios to maintain a constant leverage, which leads to a linear deleveraging rule in reaction to market shocks.

This approach was used to analyze fire-sales spillovers in the EU banking system by Green- wood et al. (2015) and in the US banking system by Duarte and Eisenbach (2013). Both studies find evidence of potentially large exposures of the banking system to contagion via fire sales. This approach has been explored by supervisors as a possible method for incorporating fire sales in macro stress tests Henry et al. (2013); Cappiello et al. (2015).

There is some empirical evidence that in the medium term large financial institutions maintain fairly stable levels of leverage Adrian and Shin (2010) but it is not clear why the same institutions would enforce such leverage targets in the short term, especially in stress scenarios where this could entail high liquidation costs.

We propose a different approach for modeling fire sales, based on the premise that deleveraging by financial institutions occurs in reaction to losses in their portfolios Kyle and Xiong (2001); Cont and Wagalath (2013). This deleveraging may be the result of investor redemptions for funds, as evidenced in Ellul et al. (2011); Coval and Stafford (2007); Shleifer and Vishny (2011); Jotikasthira et al. (2012); but for regulated financial institutions such as banks, large scale deleveraging is mainly driven by portfolio constraints – capital, leverage or liquidity constraints – which may be breached when large losses occur. We have focused for simplicity on leverage constraints, but the model is easily extendable to multiple constraints on portfolios. Given the one-sided nature of these constraints, such institutions react asymmetrically to large losses and large gains Ang

1See the Section “Process and Requirements after CCAR 2016: https://www.federalreserve.gov/

newsevents/press/bcreg/bcreg20160629a1.pdf.

et al. (2006). The asymmetry and the threshold nature of deleveraging differentiates our approach from models based on leverage targeting, leading to quite different outcomes.

Deleveraging by financial institutions impacts market prices and, when portfolios are marked to market, leads to further losses which may in turn trigger further deleverag- ing. We quantify this impact and the resulting endogenous risk, paying attention to the estimation of market impact parameters; the magnitude of these parameters, and their heterogeneity across asset classes, is shown to greatly influence the results of the stress tests.

We use these ingredients to design a systemic stress testing framework for banking systems. Application of the method to the EU banking system shows that this approach may lead to outcomes which are substantially different from single-bank stress tests. We emphasize in particular the concept of indirect exposure, which we show to be relevant for bank stress testing.

1.2 Summary and main findings

We present a framework for quantitative modeling of fire sales in a network of financial institutions with common asset holdings, subject to leverage or capital constraints. Asset sales may be triggered in reaction to external shocks to asset values when portfolios are subject to capital or leverage constraints; the market impact of these asset sales then leads to contagion of mark-to-market losses to other portfolios, which may in turn be led to deleverage if their constraints are breached. In contrast to balance sheet contagion which arises through direct bilateral exposures, this price-mediated contagion occurs through common asset holdings, even in absence of direct linkages between financial institutions.

The resulting feedback loop may lead to loss amplification, systemic risk and large-scale instability of the financial system. Our model provides a method for quantifying the exposure of the financial system to these effects, which we apply to data on European banks. This leads to several interesting findings.

• Existence of a tipping point: we show the existence of a critical macro-shock level beyond which fire sales trigger considerable contagion. The level of this critical shock depends on the institutions’ leverage, as well as the concentration, commonality and liquidity of their asset holdings. In the European banking system, this critical shock size is found to correspond to large, but not extreme, losses in asset values.

• Magnitude and heterogeneity of losses due to fire sales: we find that fire sales contribute significantly to system-wide losses in stress scenarios, accounting for more than 35% of the total losses and between 20 to 40% of system bank equity.

These results are significant enough to modify the outcome of bank stress tests.

Moreover, while the total system-wide loss can always be replicated in a stress test without fire sales by applying a larger shock to assets, the heterogeneous cross- sectional distribution of losses across banks cannot be reproduced in absence of fire sales by simply applying a larger macro-shock.

• Importance of gain-loss asymmetry and the threshold nature of fire sales:

We argue that fire sales arise when portfolio constraints such as leverage, liquidity or capital ratios are breached as a result of large portfolio losses. Theone-sidednature of these constraints leads to an asymmetric reaction of banks to gains and losses, which differentiates our model from the ‘leverage targeting’ model used in some

previous studies of fire sales Adrian and Shin (2010); Duarte and Eisenbach (2013);

Greenwood et al. (2015). Comparison with stress tests based on leverage targeting shows substantial differences in outcomes: leverage targeting models overestimate the magnitude of fire sales, especially at smaller shock levels, but underestimate the acceleration (convexity) of fire sales with increasing shock size present in the threshold model.

• Distinction between insolvency and illiquidity: unlike previous models of con- tagion, which have mainly focused on modeling insolvency, our model distinguishes between failure due to insolvency and failure due to illiquidity. We observe that, while insolvency is the dominant mode of failure of banks in scenarios associated with extremely large initial shocks, illiquidity is the dominant mode of failure in scenarios associated with moderate shocks which are nevertheless large enough to trigger fire sales.

• Indirect exposures: As a result of fire-sales spillovers, a portfolio’s exposure to an asset class in a stress scenario may be larger than its notional exposure. This naturally leads to the notion of indirect exposure, which is scenario-dependent, and can be quantified using our model. One striking finding is that many EU banks have significant (indirect) exposures to asset classes they do not hold, such as commercial and residential mortgages in EU countries where they do not issue loans.

• Sensitivity to market depth across asset classes: Calibration to data on mar- ket prices, trading volumes and turnover reveals a significant dispersion in market depth across asset classes. We show that ignoring this heterogeneity of market depth across asset classes leads to a considerable bias in the estimation of fire-sales losses in stress tests. This highlights the importance of conducting a rigorous sensitivity analysis on liquidity estimates.

• Second-round effects are significant: fire sales may lead to a feedback loop which generates market losses and further fire sales across other financial institu- tions. Ignoring these feedback effects and the corresponding second- and higher- round deleveraging may lead to a significant underestimation of system-wide losses.

These findings have many implications for risk management in financial institutions and for the monitoring systemic risk in the financial sector. In particular, they underline the need for a systemic approach to stress testing and the necessity of macroprudential tools for tackling the risks resulting from price-mediated contagion. We believe the model pre- sented here provides a useful tool for monitoring of system-wide and bank-level exposure to price-mediated contagion.

1.3 Outline

The paper is structured as follows. Section 2 introduces our modeling framework and outlines a method for systemic stress testing in presence of fire sales. Section 3 describes the application of this stress testing approach to the European banking system using data from the European Banking Authority (EBA) and details our empirical findings. In Sec- tion 4, we introduce the concept ofindirect exposureresulting from fire-sales spillovers and illustrate the magnitude of indirect exposures in the European banking system. Section 5 discusses implications of our findings for systemic stress testing, risk management in institutions and macroprudential policy.

2 Modeling spillover effects from fire sales

2.1 Balance sheets and portfolio constraints

We consider a stylized model of the financial sector, with multiple financial institutions whose portfolios may contain holdings across multiple asset classes. We will sometimes refer to these institutions as ‘banks’ although, as we will explain below, it is relevant to include non-banks in the scope of the model. Each institution holds two types of assets:

1. Illiquid assets: These are portfolio holdings which are either not easily marketable or would be subject to a deep discount if they were to be sold during a stress scenario; we assume that they are not subject to fire sales. This category includes non-securitized loans, commercial and residential mortgage exposures and retail exposures.²

2. Securities: These are positions in marketable securities which may be liquidated over a short time scale if necessary. Sovereign bonds, corporate bonds and deriva- tives exposures are included in this category. Each asset class µ in this category is characterized by a market depth parameter Dµ, whose estimation we will describe in detail below. The market liquidity of assets within this group can vary strongly and our model accounts for this heterogeneity.

Financial institutions are labeled by i = 1..N, security (types) by µ = 1..M, illiquid asset classes by κ = 1..K. Securities holdings (in euros) of institution i are denoted (Π^i,µ, µ= 1..M) and holdings in illiquid assets are denoted (Θ^i,κ, κ= 1..K).

The capital (equity) of financial institutioni is denoted Cⁱ and Ii =PK

κ=1Θ^i,κ is the total value of illiquid assets. The leverage ratio of the institution is then given by

λⁱ(Π, C, I) = PM

µ=1Π^i,µ+I_i

Cⁱ =

PM

µ=1Π^i,µ +PK κ=1Θ^i,κ

Cⁱ ≤λmax, (1)

where the upper bound λ_max > 1 corresponds to a regulatory leverage constraint, as for instance required by Basel 3. We consider here for simplicity that the leverage ratio will be the binding constraint for financial institutions’ capital, but the model may be easily adapted to include more than one constraint: for instance, if we introduce regulatory risk weights w_µ for each asset class, the constraint on the ratio of capital to risk weighted assets may be expressed as

PM

µ=1w_µΠ^i,µ+PK

κ=1w_κΘ^i,µ

Cⁱ ≤R_max.

Other portfolio constraints, such as liquidity ratios, lead to similar one-sided linear in- equalities.

The state of the financial system is summarized by the matrices of holdings Π,Θ and the capital levels C. We will now describe how this system evolves when subject to macroeconomic stress.

2This category further includes assets that do not suffer any direct or indirect losses, are not available for deleveraging, but still contribute to balance sheet size and leverage (e.g. intangible assets).

2.2 Stress scenarios

We consider stress scenarios described through the percentage loss _κ ∈ [0,100%] in values for each asset class κ, as in the approach used in regulatory bank stress tests.

For simplicity, and to emphasize contagion across asset classes, we will consider shocks to illiquid assets in the examples below, but the model perfectly accommodates stress scenarios with heterogeneous shocks across all asset classes. The initial loss of bank i in the stress scenario = (_κ, κ= 1..K) is given by

<Θⁱ, >=

K

X

κ=1

κΘ^i,κ. (2)

This results in a loss (i.e. a reduction) in the value of illiquid holdings, reducing it to I_i() :=I_i−<Θⁱ, >

and a corresponding decrease in equity:

C₀ⁱ() = (Cⁱ−<Θⁱ, >)₊. (3) The leverage of bank i then increases to

λⁱ(Π, C₀(), I()) = PM

µ=1Π^i,µ +I_i() C₀ⁱ() =

PM

µ=1Π^i,µ+I_i−<Θⁱ, >

(Cⁱ−<Θⁱ, >)₊ . (4) If this value exceeds the leverage constraintλmax then the institution needs to deleverage, i.e. sell some assets.³ This occurs if the loss level exceeds the threshold

<Θⁱ, > ≥ Cⁱ(λ_max−λⁱ(Π, C, I))

λ_max−1 , (5)

which is proportional to the capital buffer ofi or its distance from the leverage constraint prior to the shock. The amount of loss an institution can absorb before being led to deleverage is equal to its capital buffer in excess of regulatory requirements (Hellwig (2009)).

In the case where the shock is applied to a single asset classκ, institution i is led to deleverage when the (percentage) loss _κ exceeds the level

^∗_i,κ(Π, C, I) = Cⁱ(λ_max−λⁱ(Π, C, I))

(λ_max−1)Θ^i,κ . (6)

This threshold is clearly different across institutions, depending on their initial lever- age, capital buffer and holdings in the illiquid asset subject to losses. The model hence explicitly accounts for the heterogeneity in the individual bank’s resilience to losses.

Hence, any stress scenario which falls outside of the (convex) set S(Π, C,Θ) =

∈[0,1]^K

∀i= 1..N, <Θⁱ, > ≤ Cⁱ(λ_max−λⁱ(Π, C, I)) (λ_max−1) ,

. (7)

3If one allowed for negative capital by removing the positive part in (3), one would need to distinguish the additional case of negative leverage, which we avoid here. We also note that as Ii() > 0 for all reasonable shocks (c.f. Table 2), the ratio is well defined.

will lead to deleveraging by one or more financial institutions. The convex setS(Π, C,Θ) corresponds to a ‘safety zone’ of stress scenarios which all banks can withstand; the larger this safety zone, the more resilient is the network to losses.

We assume that no major deleveraging occurs inside this zone. This is different from

‘leverage targeting’ models such as Greenwood et al. (2015), where portfolios react (sym- metrically) to arbitrarily small shocks.

The size of the safety zone depends on the capital levelsC_i of the financial institutions but also on their capital buffers in excess of requirements. If the vector of shocks to assets lies outside the safety zone, then one or more institutions are led to deleverage their portfolios. We now describe how this deleveraging is modeled.

2.3 Deleveraging

If (and only if) the magnitude of losses in asset values is such that the leverage constraint is breached for institution i, it deleverages a proportion Γⁱ of its portfolio in order to restore its leverage ratio to a leverage target λ_b ≤ λ_max. This leads to the following equation for Γⁱ ∈[0,1]:

(1−Γⁱ)PM

µ=1Π^i,µ+I_i()

C₀ⁱ() =λ_b (8)

Thus, in response to an external shock , institution i needs to deleverage a fraction Γⁱ(Π, C₀()) =

PM

µ=1Π^i,µ+I_i()−λ_bC₀ⁱ() PM

µ=1Π^i,µ ∧1

! 1^PM

µ=1Πi,µ+Ii() Ci0() >λmax

(9)

= C₀ⁱ()(λⁱ(Π, C₀(), I())−λ_b) PM

µ=1Π^i,µ ∧1

!

1_λⁱ_(Π,C0(),I())>λmax,

of its marketable assets, where Π₀, C₀, I() are given by (3) In terms of theinitial capital C and the illiquid holdings Θ, this can be written as

Γⁱ(Π, C) = (λ_b−1)<Θⁱ, >+Cⁱ(λⁱ(Π, C, I)−λ_b) PM

µ=1Π^i,µ ∧1

! 1^PM

µ=1Πi,µ+I−<Θi,>

(C−<Θi,>)+ >λmax

. (10) Unlike the state variables Π, C which evolve as deleveraging occurs, Θ and (and I()), which represent respectively the holdings in illiquid assets and the initial shocks to these assets, are static parameters.

As in Greenwood et al. (2015); Duarte and Eisenbach (2013), we assume that banks delever their marketable assets proportionally. This proportional deleveraging assumption is supported by empirical studies on asset sales of large financial institutions Getmansky et al. (2016), Schaanning (2017). An alternative would be to assume a pecking order of liquidation, or that banks determine their liquidation policy by maximizing expected liquidation value Braouezec and Wagalath (2017). Greenwood et al. (2015); Duarte and Eisenbach (2013) perform robustness tests on the order of liquidation, finding that it reduces the magnitude of fire-sales losses.

Figure 1 (circles) displays the dependence of the deleveraging ratio (10) on the shock level for a portfolio holding a single class of illiquid asset: this ratio is zero for shocks lower than the threshold ^∗_i corresponding to the breach of the leverage constraint, then increases linearly thereafter. There is a discontinuity at the onset of deleveraging, which

depends on the size of the capital buffer which the bank intends to rebuild by deleveraging.

In the case where the leverage constraint is saturated after deleveraging, i.e. λ_b =λ_max, the dependence on the shock size is continuous and convex (solid line). In the example shown in Figure 1 we have assumed λ_b = 0.95λ_max, which corresponds to a safety buffer 5% below the leverage constraint.

Previous studies on fire sales have assumed instead alineardependence of the volume of deleveraging with respect to the shock size (dotted line): this is the leverage targeting model, which we will discuss further in Section 2.6.

Unlike the leverage targeting model, our model gives rise to deleveraging only if the shock level exceeds a (bank-specific) threshold ^∗_i. Once this threshold is reached, the volume of deleveraging increases linearly with (−^∗_i)₊ until no more marketable assets are available for sale (i.e. Γⁱ = 1). At this point, although the institution is still solvent, it may become illiquid. The model thus leads to a natural distinction between failures due to insolvency, which may occur if the initial loss in asset values is large enough, and failures due to illiquidity, which may occur further down the road if the liquidation of marketable assets fails to raise enough liquidity.

Figure 1: Volume of asset sales ( % of marketable assets) as a function of the percentage loss in value of illiquid assets for a portfolio with 20% in illiquid assets and initial leverage of 25, and a leverage constraint of 33. Leverage targeting leads to a linear response (dashed line), whereas our assumption of deleveraging to comply with a leverage constraint leads to no asset sales for shocks smaller than a threshold and a linear increase above the threshold (solid line). Finally, if we assume that the bank deleverages to restore a non-zero capital buffer we obtain the discontinuous response function (circles).

Summing across all institutions j = 1..N in the network yields the total volume of asset sales (in monetary units) in the stress scenario :

q^µ(,Π, C) =

N

X

j=1

Γ^j(Π, C₀())Π^j,µ

for the asset class µ. In the case λ_b = λ_max, the functions Γ^j(Π, C₀()) are convex with respect to over a large range of values, i.e. in the region max_iΓⁱ ≤ 1. In this range,

the aggregate volume of asset sales q^µ(,Π, C) exhibits a convex dependence in , which leads to a ‘multiplier effect’: as more severe stress scenarios (larger ) are considered, the marginal response in terms of deleveraging also increases. An example, based on data from the EU banking system (see next section) is shown in Figure 2.

Figure 2: Volume of asset sales ( % of marketable assets) across EU banks in reaction to a scenario in which banks realize losses of percent of the notional value on Spanish residential and commercial real estate exposures (horizontal axis). The Basel 3 leverage constraintλ_max = 33 is used in this example. The solid red line corresponds toλ_max =λ_b, the circles correspond to λ_b = 0.95λ_max.

2.4 Market impact and price-mediated contagion

If this volume of deleveraging represents a sizeable fraction of the market depth, it may have a non-negligible impact on the market price of these assets and lead to a price decline, whose magnitude ∆S^µ is an increasing function ofq^µ:

∆S^µ

S^µ =−Ψ_µ(q^µ) (11)

where Ψ_µ : R → [0,1] is a continuous, increasing and concave function with Ψ_µ(0) = 0, which we call the market impact function for asset class µ. Ψ_µ may be thought of as an inverse demand function. Assuming Ψ_µis a smooth function, linearizing for small volumes yields

Ψ_µ(q)' q

D_µ where D_µ= 1 Ψ⁰_µ(0)

is a measure ofmarket depthfor asset classµ. Naturally this quantity depends on the asset µ, as the same dollar liquidation volume q will have a different price impact depending on the asset class. A simple one-parameter specification often used in practice is Ψ_µ(q) = ψ(q/D_µ) where ψ(.) is an increasing function with ψ⁰(0) = 1. We discuss parametric specifications of Ψ_µ and their estimation in Section 3.2.

We can now describe the processes which occur at (the k-th round of) deleveraging.

Denote by Π_k−1, C_k−1, S_k−1 respectively the holdings in marketable assets, the equity and the (vector of) asset prices after k−1 rounds of deleveraging. At round k:

1. Each institution j deleverages by selling a proportion Γ^j(Πk−1, Ck−1) of its mar- ketable assets, leading to an aggregate amount q_k^µ = PN

j=1Γ^j(Π_k−1, C_k−1)Π^j,µ_k−1 of sales in asset class µ.

2. The market impact of asset sales results in a decline in market prices, moving the market price to

S_k^µ=S_k−1^µ (1−Ψ_µ(q_k^µ)). (12) 3. This decline in price changes the market value of holdings in asset classµ to

Π^i,µ_k := Π(Π_k−1, C_k−1) (13)

= 1−Γⁱ(Πk−1, Ck−1)

| {z }

Remainder after deleveraging byi

Previous value

z }| {

Π^i,µ_k−1 1−Ψ_µ

N

X

j=1

Γ^j(Πk−1, Ck−1)Π^j,µ_k−1

!!

| {z }

Price impact on remaining holdings

.

This generates two types of losses for portfolioi. First, the price moves due to the market impact of fire sales, which leads to a mark-to-market loss given by

Mⁱ(Πk−1, Ck−1) =

M

X

µ=1

(1−Γⁱ(Πk−1, Ck−1))Π^i,µ_k−1−Π^i,µ_k

(14)

= (1−Γⁱ(Πk−1, Ck−1))

M

X

µ=1

Π^i,µ_k−1Ψ_µ

N

X

j=1

Γ^j(Πk−1, Ck−1)Π^j,µ_k−1

! .

A second source of loss, not accounted for in previous studies, stems from the fact that assets are not liquidated at the current market price but at a discount: this ‘implemen- tation shortfall’, as it is called in the literature on optimal trade execution Almgren and Chriss (2000)) corresponds to the difference between the market price at the time of sale and the volume-weighted average price (VWAP) during liquidation. This VWAP lies somewhere between the pre- and post-fire-sales prices. We model it as a weighted average with weights (1−α, α), α∈ [0,1] of the pre- and post-fire-sales prices, where α = 0 cor- responds to zero implementation shortfall, and α= 1 corresponds to full implementation shortfall (assets liquidated at post-fire sales price). This leads to the general formula for the implementation shortfall:

Rⁱ(Πk−1, Ck−1) =

M

X

µ=1

Γⁱ_kΠ^i,µ_k−1− (1−α)Γⁱ_kΠ^i,µ_k−1+αΓⁱ_kΠ^i,µ_k−1(1−Ψ_µ(q_k^µ))

=αΓⁱ(Πk−1, Ck−1)

M

X

µ=1

Π^i,µ_k−1Ψ_µ

N

X

j=1

Γ^j(Πk−1, Ck−1)Π^j,µ_k−1

!

. (15)

where we wrote Γⁱ_k as shorthand for Γⁱ(Πk−1, Ck−1). In the empirical examples below, we will useα= ¹₂, which corresponds to a VWAP midway between the pre- and post-fire-sales

prices. In Cont and Schaanning (2017) it is shown that α≥ ¹₂ is a plausible assumption, also from a modeling perspective.

Summing (15) with (14) yields the total loss of portfolioi at thek-th round of devel- eraging:

Lⁱ(Π_k−1, C_k−1) =Mⁱ(Π_k−1, C_k−1) +Rⁱ(Π_k−1, C_k−1) (16)

= 1−(1−α)Γⁱ(Πk−1, Ck−1)

M

X

µ=1

Π^i,µ_k−1Ψ_µ

N

X

j=1

Γ^j(Πk−1, Ck−1)Π^j,µ_k−1

! .

This loss reduces the equity of institution iby the same amount:

C_kⁱ = C_k−1ⁱ −Lⁱ(Πk−1, Ck−1)

+ (17)

Linearizing the market impact function Ψ_µ yields Lⁱ(Πk−1, Ck−1)≈(1−(1−α)Γⁱ)

M

X

j=1 N

X

µ=1

Π^i,µ_k−1Π^j,µ_k−1 D_µ

| {z }

Ωij(Πk−1)

Γ^j = (1−(1−α)Γⁱ)

M

X

j=1

Ω_ij(Πk−1)Γ^j,

(18) which shows that the magnitude of fire-sales spillovers from institution i to institutionj is proportional to the liquidity-weighted overlap Ω_ij between portfolios i and j Cont and Wagalath (2013):

Ωij(Π) :=

M

X

µ=1

Π^i,µΠ^j,µ

D_µ . (19)

The matrix of portfolio overlaps

Ω(Π) = ΠD⁻¹Π^>, (20)

where D is the diagonal matrix of market depths D_µ, can be viewed as a weighted adja- cency matrix of the underlying network, linking portfolios through their common expo- sures. We will further analyze the properties of this matrix in Section 3.

In summary, an initial loss in asset values may trigger a feedback loop, schematically represented in Figure 3, in which, at each iteration, portfolio deleveraging leads to fire sales, leading to price declines and mark-to-market losses which may in turn trigger further fire sales. The state variables representing the matrix Π of portfolio holdings in marketable assets and the equity levels C = (C_i, i = 1..N) are initialized as described in equations (3)-(4) (for k= 0) and updated at each round of deleveraging

(Π_k, C_k) =f(Πk−1, Ck−1), (21) where Π_k is defined in (13) and

C_kⁱ = C_k−1ⁱ −Lⁱ(Πk−1, Ck−1)

+ (22)

where the loss Lⁱ is defined in (16).

Initial shock

Deleveraging

Mark to market losses

Market impact to assets

Figure 3: An initial loss in asset values may generate a feedback loop which may lead to multiple rounds of deleveraging and further declines in asset values.

2.5 Feedback loops, insolvency and illiquidity

The iteration described above continues in principle as long as at least one institution is in breach of its leverage/ capital constraint after losses due to deleveraging are accounted for. However, in the (realistic) situation where we assume that institutions build a non- zero buffer beyond the minimal capital requirements (i.e. λb < λmax in the notation of Section 2.3 ), this fire-sales cascade terminates after a finite number T of iterations Cont and Schaanning (2017). As we will discuss below, this is not the case in leverage targeting models, which lead to infinite fire-sales cascades.

Along the way, some institutions may become insolvent: this occurs if at any point in the iterations the loss Lⁱ(Π_k, C_k) exceeds the capital C_k. Then instution i becomes insolvent and does not play any further role in subsequent rounds. Another type of failure which may occur along the cascade is failure due to illiquidity: this occurs when an institution has sold all of its marketable assets and is left with no further liquid assets.

This may occur even though the institution is still solvent.

This distinction between failure due to insolvency and failure due to illiquidity is highly relevant in practice. In fact, one can note that this was precisely the scenario that occurred in the failure of Bear Stearns and Lehman Brothers.⁴ In contrast to most default risk models and previous studies on fire sales, our model distinguishes between these two causes of failure and highlights the fact that institutions can fail even when they have positive equity.

In contrast to models of default contagion, contagion of losses across institutions occurs not just at default but actually before the default of an institution, and its scope is not limited to counterparties. Deleveraging by distressed institutions, which is precisely aimed at preventingtheir default, is in fact what triggers this contagion.

Denoting byT the length of the cascade, the fire-sales loss for banki triggered by the stress scenario is given by

F Loss(i, ) =

T

X

k=1

Lⁱ(Πk−1, Ck−1). (23) and the total system-wide fire-sales loss in this scenario is

SLoss() =

N

X

i=1

F Loss(i, ). (24)

4See letter by the then SEC chairman Christopher Cox to the Basel Committee on Banking Supervision https://www.sec.gov/news/press/2008/2008-48.htm.

Note that the fire-sales loss (23) does not include the initial loss which triggers the delever- aging: in absence of deleveraging and price-mediated contagion F Loss(i) =SLoss= 0.

Table 1 summarizes the notations of the model and references the equations where they are defined.

Variable Notation Defined in

Financial institutions i, j = 1..N - Asset class: illiquid assets κ= 1..K Section 2.1 Asset class: marketable assets µ= 1..M Section 2.1

Number of iterations (rounds) k -

State variables (in EUR)

Marketable assets Π^i,µ Section 2.1

Capital Cⁱ Section 2.1

Parameters

Illiquid asset (in EUR) Θ^i,κ Section 2.1

Initial shock (in %) _κ (4)

Market depth for asset class µ Dµ (13) Key quantities

Deleveraging proportion at roundk Γⁱ_k (9)

Leverage of institution i λⁱ (1)

Fire-sales loss (k-th round) Lⁱ(Π_k−1, C_k−1) (16) Fire-sales loss for banki (all rounds) F Loss(i, ) (23) System-wide fire-sales loss SLoss() (24)

Table 1: Overview of model notations.

2.6 Comparison with “leverage targeting”

Recent empirical studies on fire-sales spillovers Duarte and Eisenbach (2013); Greenwood et al. (2015) have explored a different mechanism for fire sales, based on the idea that banks maintain a ‘leverage target’, which leads them to rebalance their portfolios in a procyclical manner following changes in asset values. An oft-cited argument to support this model is the empirical correlations between quarterly changes in asset size and debt size for banks Adrian and Shin (2010, 2014).

While both models incorporate the idea of market impact of deleveraging and the resulting endogenous portfolio losses and contagion effects, they differ in some important ways:

1. The threshold nature of fire sales: in the leverage targeting model, bank (de)leveraging in response to arbitrarily small changes in asset values, regardless of their capital or liquidity buffers, generates fire-sales losses even for low market stress levels. In our model, deleveraging only occurs when losses are large enough to trigger portfolio constraints: for shocks below this critical level, there is no deleveraging. By as- suming that all institutions constantly respond to arbitrarily small changes in asset values, the leverage targeting model overestimates the magnitude of deleveraging in response to small shocks. This is illustrated in Figure 2, which compares the over- all deleveraging across EU banks in response to losses on exposures to the Spanish housing market.

2. Dependence of deleveraging on magnitude of losses: Leverage targeting implies a volume of deleveraging linear in the size of the portfolio loss; since the volume of deleveraging is capped at 100% of assets, this leads to a concave dependence of the volume of asset sales on the shock size. By contrast, for small to moderate shocks, the volume of deleveraging has a convex dependence on the loss size in our model, as shown in Section 2.3 and illustrated in the example of Figure 2:

deleveraging accelerates as we increase the shock size to more extreme levels, leading to a ‘multiplier effect’, absent in the leverage targeting model.

3. Finite length of fire-sales cascades: The assumption of leverage targeting leads to an infinite sequence of iterations which never cease since at each round further mark-to-market losses are generated endogenously, which leads to a deviation from the target leverage and in turn generates new asset sales or purchases. In stress tests, one then needs to choose an ad-hoc number of iterations to compute the loss.

Although losses converge as we iterate this cascade, in general estimates of fire-sales losses depend on the actual number of iterations that chosen in a simulation.

By contrast, as shown in Cont and Schaanning (2017), in a threshold model with a capital buffer λ_b < λ_max, the fire-sales cascade always terminates after a finite number of iterations, typically 5 to 10 rounds in most empirical examples, as shown in the next section.

The consequences of these differences are explored in more detail in the next section, where we compare the results of stress tests performed using the two approaches, and in the companion paper Cont and Schaanning (2017). To implement the leverage targeting model in our stress test, we simply replace the deleveraging function by

Γⁱ(Π, C₀()) =

PM

µ=1Π^i,µ +Ii()−λbC₀ⁱ() PM

µ=1Π^i,µ ∧1

! .

Only marketable assets are assumed to be available for deleveraging. This assumption is different from Duarte and Eisenbach (2013); Greenwood et al. (2015), where deleveraging is applied to the entire portfolio.

3 A systemic stress test of the European banking sys- tem

We now describe how the model may be used to perform a systemic stress test, in order to quantify the exposure of the banking system to fire-sales spillovers, and apply the framework to data on the European banking system.

3.1 Data

Our empirical study is based on data from the European Banking Authority (EBA), which provides information, collected in 2011 and 2016, on notional exposures of 90 European banks across 148 asset classes.⁵ Holdings are given by asset class and geographical region.

5This dataset was also used in the study by Greenwood et al. (2015), and facilitates comparison with the literature.

Capital C

Assets

Deleveraging zone

Le ve rage con

st rai nt

No deleveraging zone

max

b

le ve rage tar ge t

I(✏)

⇧ + I (✏)

Illiquidity

In sol ve n cy

Figure 4: Evolution of asset values (vertical axis) and capital (horizontal axis) in a fire- sales cascade. The solid red line corresponds to the leverage constraint, the dashed green line denotes a target leverage corresponding to an excess capital buffer, and the dotted blue line is the path of a sample portfolio. Losses in asset values erode the equity, moving the portfolio closer to the origin; when the portfolio crosses the red line corresponding to the leverage constraint, deleveraging occurs: the institution tries to reconstitute a buffer by returning to the target leverage (dotted line). The market impact of these asset sales leads to further losses and displaces the state to the left; if it crosses the red line again, a new round of deleveraging follows etc. An institution becomes insolvent when it reaches the boundary C = 0 (vertical axis) and illiquid when it reaches the boundary Π = 0 (horizontal dashed line).

Asset classes are specified in Table 2. Greenwood et al. (2015) assumed all assets to be available for liquidation; we only consider a subset to be marketable, i.e. available for liquidation at short notice. We identify four classes of marketable assets (“securities”), which may be liquidated in a stress scenario; the other asset classes are classified as illiquid assets.⁶ Assets are further labelled by 37 geographical regions, which correspond to the 27 countries of the EU (i.e. without Croatia at the time) plus the United States (US), Norway (NO), Iceland (IS), Liechtenstein (LI), Japan (JP), Asia (A1), Other non-EEA non-emerging countries (E3), Eastern Europe non-EEA (E5), Middle and South America (M1) and Rest of the world (R5). Hence, with the four marketable asset classes and 37 geographical regions, the matrix of marketable assets Π is given by a 90×148 matrix.

6Table 10 in the Appendix, we provide the data identifiers that allow correspondence with the EBA dataset.

The illiquid asset holdings are given by a 90×75 matrix Θ. This corresponds to 74 asset classes for commercial and residential mortgage exposures respectively in the 37 regions and a 75th entry consisting of all remaining illiquid asset holdings.⁷

Illiquid assets

Residential mortgage exposures Commercial real estate exposure

Retail exposures: Revolving credits, SME, other Indirect sovereign exposures in the trading book

Defaulted exposures

Residual exposure (cf. Appendix Table 10) Securities / marketable assets

Corporate bonds Sovereign debt

Direct sovereign exposures in derivatives Institutional client exposures: interbank, CCPs,...

Table 2: Asset classes used for the stress test.

3.2 Market impact and market depth

A key assumption in models of fire-sales spillovers concerns the impact of asset liquidations on market prices. This may be summarized in the choice of a market impact function Ψ_µ, which defines the correspondence between the liquidation size q (in monetary units) and the relative price change for each asset class µ: ^∆S_Sµ^µ = −Ψ_µ(q). An adequate choice for Ψ_µ should be increasing, concave, satisfy Ψ_µ(0) = 0 and lead to non-negative prices.

Common specifications are the linear model Kyle (1985); Bertsimas and Lo (1998), Almgren and Chriss (2000); Obizhaeva (2012); Cont et al. (2014)

Ψ_µ(q) = q

D_µ with D_µ =cADV_µ

σ_µ (25)

where ADV_µ is the average daily trading volume (in EUR), σ_µ the daily volatility (in %) of the asset, ca coefficient close to 0.5, estimated from transactions data, and the square root model Bouchaud (2010)

Ψ_µ(q) =c σ_µ

r q

ADV_µ (26)

We note that both trading volume and volatility are associated with a liquidation horizon τ, taken in most studies to be daily by default. If we assume the liquidation horizon τ to be longer than a day, then the market depth parameter needs to be adjusted. In the linear impact model, the adjustment is:

D_µ(τ) =c ADV_µτ σµ

√τ =c ADV_µ σ_µ ×√

τ . (27)

7Our representation differs slightly from Greenwood et al. (2015), who considered 42 asset classes consisting of the 37 sovereign exposures by geographical region and five further classes, aggregated across all geographical regions: “commercial real estate”, “mortages”, “corporate loans”, “small and medium enterprise loans” and “retail revolving credit lines”.This leads to a less granular model compared to ours as we distinguish assets both by type and country.

This adjustment is important, and corresponds to the intuitive observation that liquidat- ing the same portfolio over a longer horizon reduces impact. The liquidation horizon τ may be interpreted as the time window the banks dispose of to comply with portfolio constraints. In the case study below we will use τ = 20 days.

By contrast, in the square root model, the impact of a transaction is invariant to a change in the liquidation horizon since the denominator and the numerator in (26) scale in the same way. This leads to the counterintuitive (and, we believe, incorrect) conclusion that impact is insensitive to the rate of liquidation. For this reason, we refrain from using the square-root model in the sequel.

In a linear impact model, asset classes are differentiated according to their market depth D_µ. Greenwood et al. (2015) assume a uniform depth D_µ = 10¹³ (EUR) for all asset classes. This homogeneity assumption is not supported by empirical studies on market impact Bouchaud (2010); Obizhaeva (2012); Cont et al. (2014), which indicate that market impact varies widely across assets. Ignoring this heterogeneity may lead to biased results, overestimating losses in more liquid asset classes while underestimating losses in less liquid asset classes.

Duarte and Eisenbach (2013) use haircuts and repo rates for determining the liquidity of different asset classes; this does introduce some heterogeneity across asset classes but the relation between these quantities and market impact is not clear. For instance, haircuts may simply reflect the volatility of an asset, rather than its liquidity or market depth.

We use a direct, data-driven approach to the modeling of market impact. To estimate the market depth parameters for each asset class using (27), we

• estimate volatility parameters σ_µ using daily returns of S&P sector indices⁸.

• obtain average daily volume estimates ADV_µ from annual volume data provided by the US Treasury and various central banks (Appendix A.1).

As noted above, we use τ = 20, which corresponds to a liquidation horizon of 4 weeks, a fairly lenient assumption. Cont and Wagalath (2016) and Obizhaeva (2012) findc≈0.33.

Ellul et al. (2011) find c≈0.2−0.3 for US corporate bonds under fire-sales pressure by insurance companies. An important difference is that in Ellul et al. (2011) the bonds are being liquidated due to the bond issuer’s credit rating being downgraded, while in our analysis, we assume that the fire sales are exogenous and not linked to the security issuers. We have used c= 0.4 here.

ADV for US corporate bonds was obtained from SIFMA (see Appendix A.1). For European bonds, we simply use the same sovereign-to-corporate ratio of ADV as ob- served in the US (567.81bn/269.8bn ≈ 0.48) to estimate the ADV of the corporate and

“institutional” asset classes for European corporate bonds.

For some asset classes, data on trading volume are unavailable (or difficult to obtain).

We work around this issue by estimating, based on OECD data, the following regression model for the relationship between average daily volume (ADV) and outstanding notional logADV_µ:=c₁log (N_µ) +c₀+ε_µ (28) whereN_µdenotes outstanding notional, and using it to estimate volume for the remaining asset classes. Table 3 shows the results of this regression analysis.

8 http://us.spindices.com/indices/fixed-income/sp-eurozone-sovereign-bond-index http://us.spindices.com/index-family/us-treasury-and-us-agency/all

http://us.spindices.com/indices/fixed-income/sp-500-investment-grade-corporate-bond-index

Coefficient US treas US corp UK DE ES c₁: 0.53*** 0.64*** 0.56*** 0.48 0.94***

std. dev. 0.13 0.15 0.05 0.30 0.14

c0: 0.63 -1.1** -0.35** 0.4 -1.35***

std. dev. 0.55 0.55 0.14 0.99 0.37

adj. R² 0.39 0.60 0.93 0.15 0.76

n 19 19 11 10 15

Table 3: Logarithmic regression of bond trading volume on outstanding notional (Equa- tion 28).

Figure 5 shows the distribution of market depth estimates for all asset classes in the EBA dataset, on a logarithmic scale. The histogram reveals considerable heterogeneity in the cross-sectional distribution of market depth, with 4 orders of magnitude separating the most liquid from the least liquid assets.

Market depth (EUR) Percent 0.00.20.40.6

10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵

Holdings (EUR)

Market depth (EUR)

10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹² 109 1010 1011 1012 1013 1014 1015

Figure 5: Left: Histogram of estimated market depths for the 148 asset classes in the EBA dataset. The dashed vertical line indicates the holdings-weighted average depth given by (31).Right: Scatter plot of depth vs. holdings.

Table 4 provides values for average daily volumes (ADV) and market depths for the asset classes representing the largest holdings. These values are used as base values; we will later perform a sensitivity analysis of our results with respect to changes in these values.

Extrapolation to large volumes When applied to large transaction volumes, linear or square-root impact models may lead to negative prices. In Greenwood et al. (2015) this was addressed by capping the loss at 100%: Ψ_µ(q) = minn

1,_D^q

µ

o

. Cifuentes et al.

(2005) use an exponential specification

Ψµ(q) = 1−exp

− q D_µ

(29) which also ensures that prices remain non-negative but gives a concave impact. Never- theless, prices can get arbitrarily close to zero in both of these models. However, it is

Asset class ADV Market depthDµ Impact of 10 bn (sovereign unless

specified)

(bn EUR) (10¹² EUR)

US 567.8 428.8 0.2332

US (corp.) 269.8 70.2 1.424

IT 28.82 21.8 4.585

ES 28.0 21.1 4.737

DE 24.7 18.7 5.345

GB 24 18.1 5.522

FR 10 7.58 13.18

GR 8.17 6.18 16.16

SE 3.92 2.96 33.67

PT 2.27 1.71 58.14

Table 4: Average daily trading volume and estimated market depth over τ = 20 days, for the largest holdings in the EBA dataset. The impact in basis points are also given for a liquidation of 10 bn EUR over τ = 20 days.

realistic to assume that long before the price level reaches zero, arbitrageurs will step in to purchase assets subject to fire sales at a discount Shleifer and Vishny (1992). To capture this effect we introduce a price floorB_µ >0 and consider a two-parameter level-dependent price impact function:

Ψ_µ(q, S) :=

1− B_µ

S 1−exp(−q δ_µ)

. (30)

B_µ determines how far the price can fall in a fire-sales scenario. Note that this is a lower bound for the price and in a given stress test the price may not actually fall to this level.

In the empirical examples below, we set B_µ at 50 % of the market price levels. Choosing δµ =

1− B_µ S₀^µ

Dµ

makes the specification (30) compatible with the linear specification (25) for small vol- umes.

3.3 Portfolio overlaps

As shown in Eq. (18), the transmission of fire-sales losses from portfolio i to j depends on the liquidity-weighted overlap between portfolio i and j:

Ω_ij(Π) :=

M

X

µ=1

Π^i,µΠ^j,µ D_µ .

The matrix Ω of liquidity-weighted overlaps thus plays an important role in the trans- mission of fire-sales losses, which may be viewed as a contagion process on a network of financial institutions in which the link from i to j is weighted according to the liquidity- weighted overlap Ω_ij. We call this network the indirect contagion network. Similar net- work structures were explored by Braverman and Minca (2016); Guo et al. (2015) for

mutual funds. Figure 6 displays the indirect contagion network for European banks as implied by EBA data collected in 2011. The nodes correspond to different banks, with node size proportional to balance sheet size, and edges correspond to non-zero portfolio overlaps, with edge widths proportional to the logarithm of the liquidity-weighted overlap Ω_ij.

Figure 7 (left) shows the distribution of Ω_ij in this network. The peak at zero reflects the fact that many pairs of banks have no common asset holdings (so, zero overlap), i.e.

the network is sparse. On the other hand, the values are dispersed over five orders of magnitude, which illustrates the heterogeneity of the network.

Figure 6: The core of the European indirect contagion network: Node sizes are pro- portional to balance sheet size. Edge widths are proportional to the liquidity-weighted overlap. Red nodes correspond to the banks with highest loading in the first principal component of the portfolio overlap matrix Ω.

The overlap matrix Ω also gives a glimpse of the nature of ‘second-round’ contagion effects in this network. The element (i, j) of the matrix Ω² may be interpreted as the